\(\int \frac {450 e^{4 x}+e^x x^6+e^{5 x} (5625-450 e x+9 e^2 x^2)+e^{2 x} (12 x^3-12 x^4)+e^{3 x} (-150 x^3+6 e x^4)}{x^6+e^{4 x} (5625-450 e x+9 e^2 x^2)+e^{2 x} (-150 x^3+6 e x^4)} \, dx\) [7938]

Optimal result

Integrand size = 115, antiderivative size = 27 \[ \int \frac {450 e^{4 x}+e^x x^6+e^{5 x} \left (5625-450 e x+9 e^2 x^2\right )+e^{2 x} \left (12 x^3-12 x^4\right )+e^{3 x} \left (-150 x^3+6 e x^4\right )}{x^6+e^{4 x} \left (5625-450 e x+9 e^2 x^2\right )+e^{2 x} \left (-150 x^3+6 e x^4\right )} \, dx=e^x-\frac {2}{e-\frac {25}{x}+\frac {1}{3} e^{-2 x} x^2} \]

[Out]

exp(x)-2/(1/3*x^2/exp(x)^2-25/x+exp(1))

Rubi [F]

\[ \int \frac {450 e^{4 x}+e^x x^6+e^{5 x} \left (5625-450 e x+9 e^2 x^2\right )+e^{2 x} \left (12 x^3-12 x^4\right )+e^{3 x} \left (-150 x^3+6 e x^4\right )}{x^6+e^{4 x} \left (5625-450 e x+9 e^2 x^2\right )+e^{2 x} \left (-150 x^3+6 e x^4\right )} \, dx=\int \frac {450 e^{4 x}+e^x x^6+e^{5 x} \left (5625-450 e x+9 e^2 x^2\right )+e^{2 x} \left (12 x^3-12 x^4\right )+e^{3 x} \left (-150 x^3+6 e x^4\right )}{x^6+e^{4 x} \left (5625-450 e x+9 e^2 x^2\right )+e^{2 x} \left (-150 x^3+6 e x^4\right )} \, dx \]

[In]

Int[(450*E^(4*x) + E^x*x^6 + E^(5*x)*(5625 - 450*E*x + 9*E^2*x^2) + E^(2*x)*(12*x^3 - 12*x^4) + E^(3*x)*(-150*
x^3 + 6*E*x^4))/(x^6 + E^(4*x)*(5625 - 450*E*x + 9*E^2*x^2) + E^(2*x)*(-150*x^3 + 6*E*x^4)),x]

[Out]

E^x + 50/(E*(25 - E*x)) + (19531250*(50 + 3*E)*Defer[Int][(-75*E^(2*x) + 3*E^(1 + 2*x)*x + x^3)^(-2), x])/E^7
+ (1562500*(25 + E)*Defer[Int][x/(-75*E^(2*x) + 3*E^(1 + 2*x)*x + x^3)^2, x])/E^6 + (31250*(50 + E)*Defer[Int]
[x^2/(-75*E^(2*x) + 3*E^(1 + 2*x)*x + x^3)^2, x])/E^5 + (62500*Defer[Int][x^3/(-75*E^(2*x) + 3*E^(1 + 2*x)*x +
x^3)^2, x])/E^4 + (50*(50 - E)*Defer[Int][x^4/(-75*E^(2*x) + 3*E^(1 + 2*x)*x + x^3)^2, x])/E^3 + (4*(25 - E)*
Defer[Int][x^5/(-75*E^(2*x) + 3*E^(1 + 2*x)*x + x^3)^2, x])/E^2 + (4*Defer[Int][x^6/(-75*E^(2*x) + 3*E^(1 + 2*
x)*x + x^3)^2, x])/E + (12207031250*Defer[Int][1/((-25 + E*x)^2*(-75*E^(2*x) + 3*E^(1 + 2*x)*x + x^3)^2), x])/
E^6 + (976562500*(25 + 2*E)*Defer[Int][1/((-25 + E*x)*(-75*E^(2*x) + 3*E^(1 + 2*x)*x + x^3)^2), x])/E^7 - (250
0*(25 + E)*Defer[Int][(-75*E^(2*x) + 3*E^(1 + 2*x)*x + x^3)^(-1), x])/E^4 - (2500*Defer[Int][x/(-75*E^(2*x) +
3*E^(1 + 2*x)*x + x^3), x])/E^3 - (4*(25 - E)*Defer[Int][x^2/(-75*E^(2*x) + 3*E^(1 + 2*x)*x + x^3), x])/E^2 -
(4*Defer[Int][x^3/(-75*E^(2*x) + 3*E^(1 + 2*x)*x + x^3), x])/E - (1562500*Defer[Int][1/((-25 + E*x)^2*(-75*E^(
2*x) + 3*E^(1 + 2*x)*x + x^3)), x])/E^3 - (62500*(25 + 2*E)*Defer[Int][1/((-25 + E*x)*(-75*E^(2*x) + 3*E^(1 +
2*x)*x + x^3)), x])/E^4

Rubi steps \begin{align*} \text {integral}& = \int \frac {450 e^{4 x}+e^x x^6+e^{5 x} \left (5625-450 e x+9 e^2 x^2\right )+e^{2 x} \left (12 x^3-12 x^4\right )+e^{3 x} \left (-150 x^3+6 e x^4\right )}{\left (75 e^{2 x}-3 e^{1+2 x} x-x^3\right )^2} \, dx \\ & = \int \left (e^x+\frac {50}{(-25+e x)^2}+\frac {2 x^6 \left (75-2 (25+e) x+2 e x^2\right )}{(25-e x)^2 \left (75 e^{2 x}-3 e^{1+2 x} x-x^3\right )^2}+\frac {4 x^3 \left (50-(25+e) x+e x^2\right )}{(25-e x)^2 \left (75 e^{2 x}-3 e^{1+2 x} x-x^3\right )}\right ) \, dx \\ & = \frac {50}{e (25-e x)}+2 \int \frac {x^6 \left (75-2 (25+e) x+2 e x^2\right )}{(25-e x)^2 \left (75 e^{2 x}-3 e^{1+2 x} x-x^3\right )^2} \, dx+4 \int \frac {x^3 \left (50-(25+e) x+e x^2\right )}{(25-e x)^2 \left (75 e^{2 x}-3 e^{1+2 x} x-x^3\right )} \, dx+\int e^x \, dx \\ & = e^x+\frac {50}{e (25-e x)}+2 \int \left (\frac {9765625 (50+3 e)}{e^7 \left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )^2}+\frac {781250 (25+e) x}{e^6 \left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )^2}+\frac {15625 (50+e) x^2}{e^5 \left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )^2}+\frac {31250 x^3}{e^4 \left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )^2}-\frac {25 (-50+e) x^4}{e^3 \left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )^2}-\frac {2 (-25+e) x^5}{e^2 \left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )^2}+\frac {2 x^6}{e \left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )^2}+\frac {6103515625}{e^6 (-25+e x)^2 \left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )^2}+\frac {488281250 (25+2 e)}{e^7 (-25+e x) \left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )^2}\right ) \, dx+4 \int \left (-\frac {625 (25+e)}{e^4 \left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )}-\frac {625 x}{e^3 \left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )}+\frac {(-25+e) x^2}{e^2 \left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )}-\frac {x^3}{e \left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )}-\frac {390625}{e^3 (-25+e x)^2 \left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )}-\frac {15625 (25+2 e)}{e^4 (-25+e x) \left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )}\right ) \, dx \\ & = e^x+\frac {50}{e (25-e x)}+\frac {12207031250 \int \frac {1}{(-25+e x)^2 \left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )^2} \, dx}{e^6}+\frac {62500 \int \frac {x^3}{\left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )^2} \, dx}{e^4}-\frac {2500 \int \frac {x}{-75 e^{2 x}+3 e^{1+2 x} x+x^3} \, dx}{e^3}-\frac {1562500 \int \frac {1}{(-25+e x)^2 \left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )} \, dx}{e^3}+\frac {(50 (50-e)) \int \frac {x^4}{\left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )^2} \, dx}{e^3}+\frac {(4 (25-e)) \int \frac {x^5}{\left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )^2} \, dx}{e^2}-\frac {(4 (25-e)) \int \frac {x^2}{-75 e^{2 x}+3 e^{1+2 x} x+x^3} \, dx}{e^2}+\frac {4 \int \frac {x^6}{\left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )^2} \, dx}{e}-\frac {4 \int \frac {x^3}{-75 e^{2 x}+3 e^{1+2 x} x+x^3} \, dx}{e}+\frac {(1562500 (25+e)) \int \frac {x}{\left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )^2} \, dx}{e^6}-\frac {(2500 (25+e)) \int \frac {1}{-75 e^{2 x}+3 e^{1+2 x} x+x^3} \, dx}{e^4}+\frac {(31250 (50+e)) \int \frac {x^2}{\left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )^2} \, dx}{e^5}+\frac {(976562500 (25+2 e)) \int \frac {1}{(-25+e x) \left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )^2} \, dx}{e^7}-\frac {(62500 (25+2 e)) \int \frac {1}{(-25+e x) \left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )} \, dx}{e^4}+\frac {(19531250 (50+3 e)) \int \frac {1}{\left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )^2} \, dx}{e^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89 \[ \int \frac {450 e^{4 x}+e^x x^6+e^{5 x} \left (5625-450 e x+9 e^2 x^2\right )+e^{2 x} \left (12 x^3-12 x^4\right )+e^{3 x} \left (-150 x^3+6 e x^4\right )}{x^6+e^{4 x} \left (5625-450 e x+9 e^2 x^2\right )+e^{2 x} \left (-150 x^3+6 e x^4\right )} \, dx=e^x-\frac {50}{e (-25+e x)}+\frac {2 x^4}{(-25+e x) \left (-75 e^{2 x}+3 e^{1+2 x} x+x^3\right )} \]

[In]

Integrate[(450*E^(4*x) + E^x*x^6 + E^(5*x)*(5625 - 450*E*x + 9*E^2*x^2) + E^(2*x)*(12*x^3 - 12*x^4) + E^(3*x)*
(-150*x^3 + 6*E*x^4))/(x^6 + E^(4*x)*(5625 - 450*E*x + 9*E^2*x^2) + E^(2*x)*(-150*x^3 + 6*E*x^4)),x]

[Out]

E^x - 50/(E*(-25 + E*x)) + (2*x^4)/((-25 + E*x)*(-75*E^(2*x) + 3*E^(1 + 2*x)*x + x^3))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(49\) vs. \(2(24)=48\).

Time = 0.17 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.85

method	result	size
risch	\(-\frac {50 \,{\mathrm e}^{-1}}{x \,{\mathrm e}-25}+{\mathrm e}^{x}+\frac {2 x^{4}}{\left (x \,{\mathrm e}-25\right ) \left (3 x \,{\mathrm e}^{1+2 x}+x^{3}-75 \,{\mathrm e}^{2 x}\right )}\)	\(50\)
parallelrisch	\(\frac {3 \,{\mathrm e} \,{\mathrm e}^{3 x} x +{\mathrm e}^{x} x^{3}-6 x \,{\mathrm e}^{2 x}-75 \,{\mathrm e}^{3 x}}{3 x \,{\mathrm e} \,{\mathrm e}^{2 x}+x^{3}-75 \,{\mathrm e}^{2 x}}\)	\(52\)

[In]

int(((9*x^2*exp(1)^2-450*x*exp(1)+5625)*exp(x)^5+450*exp(x)^4+(6*x^4*exp(1)-150*x^3)*exp(x)^3+(-12*x^4+12*x^3)
*exp(x)^2+x^6*exp(x))/((9*x^2*exp(1)^2-450*x*exp(1)+5625)*exp(x)^4+(6*x^4*exp(1)-150*x^3)*exp(x)^2+x^6),x,meth
od=_RETURNVERBOSE)

[Out]

-50*exp(-1)/(x*exp(1)-25)+exp(x)+2*x^4/(x*exp(1)-25)/(3*x*exp(1+2*x)+x^3-75*exp(2*x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (25) = 50\).

Time = 0.25 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22 \[ \int \frac {450 e^{4 x}+e^x x^6+e^{5 x} \left (5625-450 e x+9 e^2 x^2\right )+e^{2 x} \left (12 x^3-12 x^4\right )+e^{3 x} \left (-150 x^3+6 e x^4\right )}{x^6+e^{4 x} \left (5625-450 e x+9 e^2 x^2\right )+e^{2 x} \left (-150 x^3+6 e x^4\right )} \, dx=\frac {x^{3} e^{\left (x + 1\right )} + 2 \, x^{3} + 3 \, {\left (x e^{2} - 25 \, e\right )} e^{\left (3 \, x\right )} - 150 \, e^{\left (2 \, x\right )}}{x^{3} e + 3 \, {\left (x e^{2} - 25 \, e\right )} e^{\left (2 \, x\right )}} \]

[In]

integrate(((9*x^2*exp(1)^2-450*x*exp(1)+5625)*exp(x)^5+450*exp(x)^4+(6*x^4*exp(1)-150*x^3)*exp(x)^3+(-12*x^4+1
2*x^3)*exp(x)^2+x^6*exp(x))/((9*x^2*exp(1)^2-450*x*exp(1)+5625)*exp(x)^4+(6*x^4*exp(1)-150*x^3)*exp(x)^2+x^6),
x, algorithm="fricas")

[Out]

(x^3*e^(x + 1) + 2*x^3 + 3*(x*e^2 - 25*e)*e^(3*x) - 150*e^(2*x))/(x^3*e + 3*(x*e^2 - 25*e)*e^(2*x))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (20) = 40\).

Time = 0.17 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.96 \[ \int \frac {450 e^{4 x}+e^x x^6+e^{5 x} \left (5625-450 e x+9 e^2 x^2\right )+e^{2 x} \left (12 x^3-12 x^4\right )+e^{3 x} \left (-150 x^3+6 e x^4\right )}{x^6+e^{4 x} \left (5625-450 e x+9 e^2 x^2\right )+e^{2 x} \left (-150 x^3+6 e x^4\right )} \, dx=\frac {2 x^{4}}{e x^{4} - 25 x^{3} + \left (3 x^{2} e^{2} - 150 e x + 1875\right ) e^{2 x}} + e^{x} - \frac {50}{x e^{2} - 25 e} \]

[In]

integrate(((9*x**2*exp(1)**2-450*x*exp(1)+5625)*exp(x)**5+450*exp(x)**4+(6*x**4*exp(1)-150*x**3)*exp(x)**3+(-1
2*x**4+12*x**3)*exp(x)**2+x**6*exp(x))/((9*x**2*exp(1)**2-450*x*exp(1)+5625)*exp(x)**4+(6*x**4*exp(1)-150*x**3
)*exp(x)**2+x**6),x)

[Out]

2*x**4/(E*x**4 - 25*x**3 + (3*x**2*exp(2) - 150*E*x + 1875)*exp(2*x)) + exp(x) - 50/(x*exp(2) - 25*E)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (25) = 50\).

Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22 \[ \int \frac {450 e^{4 x}+e^x x^6+e^{5 x} \left (5625-450 e x+9 e^2 x^2\right )+e^{2 x} \left (12 x^3-12 x^4\right )+e^{3 x} \left (-150 x^3+6 e x^4\right )}{x^6+e^{4 x} \left (5625-450 e x+9 e^2 x^2\right )+e^{2 x} \left (-150 x^3+6 e x^4\right )} \, dx=\frac {x^{3} e^{\left (x + 1\right )} + 2 \, x^{3} + 3 \, {\left (x e^{2} - 25 \, e\right )} e^{\left (3 \, x\right )} - 150 \, e^{\left (2 \, x\right )}}{x^{3} e + 3 \, {\left (x e^{2} - 25 \, e\right )} e^{\left (2 \, x\right )}} \]

[In]

integrate(((9*x^2*exp(1)^2-450*x*exp(1)+5625)*exp(x)^5+450*exp(x)^4+(6*x^4*exp(1)-150*x^3)*exp(x)^3+(-12*x^4+1
2*x^3)*exp(x)^2+x^6*exp(x))/((9*x^2*exp(1)^2-450*x*exp(1)+5625)*exp(x)^4+(6*x^4*exp(1)-150*x^3)*exp(x)^2+x^6),
x, algorithm="maxima")

[Out]

(x^3*e^(x + 1) + 2*x^3 + 3*(x*e^2 - 25*e)*e^(3*x) - 150*e^(2*x))/(x^3*e + 3*(x*e^2 - 25*e)*e^(2*x))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (25) = 50\).

Time = 0.91 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.37 \[ \int \frac {450 e^{4 x}+e^x x^6+e^{5 x} \left (5625-450 e x+9 e^2 x^2\right )+e^{2 x} \left (12 x^3-12 x^4\right )+e^{3 x} \left (-150 x^3+6 e x^4\right )}{x^6+e^{4 x} \left (5625-450 e x+9 e^2 x^2\right )+e^{2 x} \left (-150 x^3+6 e x^4\right )} \, dx=\frac {x^{3} e^{\left (x + 1\right )} + 2 \, x^{3} + 3 \, x e^{\left (3 \, x + 2\right )} - 150 \, e^{\left (2 \, x\right )} - 75 \, e^{\left (3 \, x + 1\right )}}{x^{3} e + 3 \, x e^{\left (2 \, x + 2\right )} - 75 \, e^{\left (2 \, x + 1\right )}} \]

[In]

integrate(((9*x^2*exp(1)^2-450*x*exp(1)+5625)*exp(x)^5+450*exp(x)^4+(6*x^4*exp(1)-150*x^3)*exp(x)^3+(-12*x^4+1
2*x^3)*exp(x)^2+x^6*exp(x))/((9*x^2*exp(1)^2-450*x*exp(1)+5625)*exp(x)^4+(6*x^4*exp(1)-150*x^3)*exp(x)^2+x^6),
x, algorithm="giac")

[Out]

(x^3*e^(x + 1) + 2*x^3 + 3*x*e^(3*x + 2) - 150*e^(2*x) - 75*e^(3*x + 1))/(x^3*e + 3*x*e^(2*x + 2) - 75*e^(2*x
+ 1))

Mupad [B] (verification not implemented)

Time = 0.57 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.26 \[ \int \frac {450 e^{4 x}+e^x x^6+e^{5 x} \left (5625-450 e x+9 e^2 x^2\right )+e^{2 x} \left (12 x^3-12 x^4\right )+e^{3 x} \left (-150 x^3+6 e x^4\right )}{x^6+e^{4 x} \left (5625-450 e x+9 e^2 x^2\right )+e^{2 x} \left (-150 x^3+6 e x^4\right )} \, dx=\frac {{\mathrm {e}}^{-1}\,\left (2\,x^3-75\,{\mathrm {e}}^{3\,x}\,\mathrm {e}-150\,{\mathrm {e}}^{2\,x}+3\,x\,{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^2+x^3\,\mathrm {e}\,{\mathrm {e}}^x\right )}{x^3-75\,{\mathrm {e}}^{2\,x}+3\,x\,{\mathrm {e}}^{2\,x}\,\mathrm {e}} \]

[In]

int((450*exp(4*x) + x^6*exp(x) + exp(2*x)*(12*x^3 - 12*x^4) + exp(3*x)*(6*x^4*exp(1) - 150*x^3) + exp(5*x)*(9*
x^2*exp(2) - 450*x*exp(1) + 5625))/(exp(2*x)*(6*x^4*exp(1) - 150*x^3) + exp(4*x)*(9*x^2*exp(2) - 450*x*exp(1)
+ 5625) + x^6),x)

[Out]

(exp(-1)*(2*x^3 - 75*exp(3*x)*exp(1) - 150*exp(2*x) + 3*x*exp(3*x)*exp(2) + x^3*exp(1)*exp(x)))/(x^3 - 75*exp(
2*x) + 3*x*exp(2*x)*exp(1))